From c81224d1705f27b84efb5225b9e1069db7fa46d7 Mon Sep 17 00:00:00 2001 From: Joe Corneli Date: Sat, 2 Sep 2017 18:18:13 +0100 Subject: [PATCH] update content --- latex/arxana-merge.tex | 11 +- org/lightning.sty | 19 ++ org/mom.png | Bin 0 -> 106810 bytes org/scheme-talk.org | 602 +++++++++++++++++++------------------------------ 4 files changed, 263 insertions(+), 369 deletions(-) create mode 100644 org/lightning.sty create mode 100644 org/mom.png rewrite org/scheme-talk.org (72%) diff --git a/latex/arxana-merge.tex b/latex/arxana-merge.tex index 242f0d8..6fcbda0 100644 --- a/latex/arxana-merge.tex +++ b/latex/arxana-merge.tex @@ -2483,7 +2483,7 @@ We will create a new plexus to store this content. (defvar iatc-intermediate-1.5 '(easy)) (defvar iatc-intermediate-1 '(not goal plausible beautiful useful heuristic)) -(defvar iatc-intermediate-A '(and)) +(defvar iatc-intermediate-A '(conjunction and)) (defvar iatc-intermediate-1A '(case_split)) (defvar iatc-content-2 '(used_in sub_prop reform instantiates @@ -2517,9 +2517,6 @@ mapped into the graph. \end{notate} \begin{elisp} -(defun highlight-tree-wrapper (tree) - (second (highlight-tree tree))) - (defun sexps-to-subgraph (tree &rest more) ;; This function will take in a tree (or maybe several) that ;; indicate a given subgraph, mark this subgraph with a cone, @@ -2548,10 +2545,16 @@ mapped into the graph. ; (get-verbs '(a (b (c 1 2)) 8 9 (d (e 1 2 3) (g 4 5)))) +(defun highlight-tree-wrapper (tree) + (second (highlight-tree tree))) (defun highlight-tree (tree &optional highlights) (with-current-plexus (second plexus-registry) + ;; Cases: + ;; IATC - create an internal identifier for it. + ;; generic hcode - store the whole form as the contents of a node. + ;; already-imported IATC - don't import it again, just refer to it (if (member (car tree) iatc-verbs) ;; yes the verb is an iatc-verb: we need to import the tree -> (let* ((verb (car tree)) diff --git a/org/lightning.sty b/org/lightning.sty new file mode 100644 index 0000000..e734dd0 --- /dev/null +++ b/org/lightning.sty @@ -0,0 +1,19 @@ +\usepackage{tikz} + +\newcommand*{\Lightning}[1][]{% + \tikz[ + x=.55 * height("H"), % symbol width + y=height("H"), % symbol height + baseline=(current bounding box.south), + line width=.02 * height("H"), + line join=bevel, + ] + \filldraw[{#1}] + (-.5, -.5) -- (.4, -.03) -- (-.1, .06) -- + (.5, .5) -- (-.4, .03) -- (.1, -.06) -- + cycle + % side bearings + (-.5 - .08, 0) + (.5 + .08, 0) + ;% +} diff --git a/org/mom.png b/org/mom.png new file mode 100644 index 0000000000000000000000000000000000000000..d77b121848114d9800cc599ad9517ece198a5bb7 GIT binary patch literal 106810 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(Scheme@ICFP) -#+DESCRIPTION: Organizer for presentation on arxana and math text analysis at Oxford. -#+KEYWORDS: arxana, hypertext, inference anchoring -#+LANGUAGE: en -#+OPTIONS: H:2 num:t toc:t \n:nil @:t ::t |:t ^:nil -:t f:t *:t <:t -#+OPTIONS: TeX:t LaTeX:nil skip:nil d:nil todo:t pri:nil tags:not-in-toc -#+INFOJS_OPT: view:nil toc:nil ltoc:t mouse:underline buttons:0 path:http://orgmode.org/org-info.js -#+EXPORT_SELECT_TAGS: export -#+EXPORT_EXCLUDE_TAGS: noexport -#+LINK_UP: -#+LINK_HOME: -#+HTML_HEAD: -#+LATEX_HEADER: \usepackage[backend=bibtex,sorting=none]{biblatex} -#+LATEX_HEADER: \bibliography{arxana-redux-refs} -#+LATEX_HEADER: \beamertemplatenavigationsymbolsempty -#+STARTUP: showeverything - -* Title slide - -Extending the LISP model: -from cons cells to triples, from trees to hypergraphs - -Joseph Corneli and Raymond Puzio - -Saturday 3 September, 2017 -Scheme@ICFP -Oxford, UK -* Overview 1: From cons cells to triples - -*Arxana* uses a higher-dimensional variant of LISP, based on nested -semantic networks instead of cons cells. In contradistinction to LISP -where the fundamental building block is a cell =(a . b)=, Arxana's -fundamental building block is a triple, =(a c b)=. - -Instead of =car= and =cdr=, we have =src=, =txt=, and =snk=. - -* Overview 2: From trees to hypergraphs - -Furthermore, in the language of the Semantic Web, every triple is -`reified'. Links themselves, as well as their constituent positions, -can be augmented with offset annotations, and can contain further -structure. - -Pierre Delacaze shared an illustrative example along these lines: - -/Mom resents the fact that John disapproves of Jane and Jim's -marriage./ - -* Overview 3: Illustration of basic mechanics - -"Nodes and links" are all represented as triples, which we call -/nemas/. A repository of nemas is a /plexus/. - -1. Each plexus has a distinguished element =0=, called *ground*. - -2. If we set =α:=(a d e)=, then =(α c b)= is an assertion *about* the - link =(a d e)=. - -3. On the other hand, =((get-src α) d e)= about the *src* of =α=, - which happens to be =a= at present, although nemas are mutable. - -4. If we wanted to make an assertion about =a= "itself", we can - represent it as a nema with the special form =β:=(0 a - 0)=. Subsequently, =(β c b)= is an assertion about =a=. - -[That said, =β= would still be mutable, is this a problem? -JC] -* Overview 4: What do we get? - -These facilities allow us to build, reason about, query, and program -with hypergraphs rather than trees or networks. - -Unlike other triple systems, code can be stored on links. This -representation strategy is useful for building runnable conceptual -models of complex and recursive structures. - -In particular, we have found this representation strategy well-suited -to modelling the non-deductive style of informal mathematical -dialogues and expositions. - -* Overview 5: Related work - -Historical programming languages which support a similar annotative -style include Kurzweil el al's /Flare/ and Nelson's /ZigZag/. - -A more recent endeavour is /AtomSpace/, which is a more mature piece -of software, e.g., it comes with both Scheme and Python bindings and a -range of optimisations. - -The relative simplicity of Arxana makes it useful for prototyping! -Also, it is implemented in Emacs Lisp, which we have found useful (in -particular, stable) over a long and slow development cycle. - -* History of development and outreach - -- Free Culture and the Digital Library Symposium, Atlanta, 2005 -- Experiments with a Common Lisp + cl-elephant backend, 2007 -- Rewrite with a Common Lisp + SQL backend, 2010 -- Rewrite with the storage model described here, 2013 -- Presentations at LispNYC, 2005 and 2013 -- Presentation at the Emacs Conference, London, 2013 -- Presentation at the monthly Emacs Meetup, NYC, 2017 -- Next Saturday: presentation at FARM 2017 - -* Illustration: Application to mathematics -``I have seen this problem, that if $G$ is a finite group and $H$ is a -proper subgroup of $G$ with finite index then $ G \neq -\bigcup\limits_{g \in G} ghg^{-1}$. Does this remain true for the -infinite case also.'' -#+ATTR_ORG: :width 800 -#+ATTR_LATEX: :width 11cm - [[file:./example-graph.png]] -*``There's something I don't understand here: do you perhaps mean -$gHg^{-1}$ instead of $ghg^{-1}$?''* - -** Relevant prior work: Conceptual Dependence theory - -E.g., "John gave Mary a book." -#+ATTR_ORG: :width 800 -#+ATTR_LATEX: :width 11cm - [[file:./john-example.png]] -#+BEGIN_SRC lisp -(atrans actor (person name john) - object (phys-obj type book) - recip (person name mary)) -#+END_SRC - -\bigskip - -*NB. We view Conceptual Dependence /primitives/ as analogous to IAT -/performatives/.* - -* The end - -That's all for now! - -- joseph.corneli@ed.ac.uk -- rspuzio@planetmath.info - -* outtakes :noexport: - -** Other things we've worked on - -- "nemas" document: http://metameso.org/ar/metamathematics.pdf -- Joe's thesis Appendix B: http://metameso.org/ar/thesis-appendix-b.pdf -- Metamathematics preprint: http://metameso.org/ar/mathematical-semantics.pdf -- hcode function spec: http://metameso.org/ar/hcode.pdf -- Lakatos game spec: http://metameso.org/ar/lakatos-flowchart.pdf -- [[http://metameso.org/cgi-bin/current.pl?action=browse;oldid=Arxana;id=A_scholium-based_document_model][A scholium-based document model]] is the wiki homepage for Arxana and has several relevant discussions. -- [[http://metameso.org/cgi-bin/current.pl/Two_years_later][Two years later]] collects many of the mathematical themes in - overview -- [[http://metameso.org/cgi-bin/current.pl/samples_of_formal_mathematical_writing][samples of formal mathematical writing]] is a small collection - of statements in mathematical language (leaving out ``the part - in between dollar signs'') - - -** Infrastructure - -This file is in versioned storage in the "mob" branch of our arxana -repo; instructions to access are here: -http://repo.or.cz/w/arxana.git/blob_plain/d187e244a8eb7d2208f1fe98db1ef895c6cd4a36:/README.mob - -A working copy of this repository is checked out at -=/home/shared-folder/arxana-redux/arxana/= on the Linode. - -And a "scratch" copy of the file has been copied to: -=/home/shared-folder/arxana-redux/arxana-redux.org= - -This can be edited in live sessions using =togetherly.el=. - -** Progress on paper - -*** Corpus/Examples: Hand-drawn diagrams in IAT+C - -/We are allowed to submit an appendix without a size limit, so if we don't have room for all of the diagrams in the paper, we could put them in the appendix./ - -- Minipolymath, treated by Gabi in the Transactions on Social Computing paper, pdf available from https://www.overleaf.com/read/jyhzpqsythzq -- Our randomly selected MathOverflow discussion (from Q&A article) -- Gowers 2012 trick problem, http://metameso.org/ar/gowers-example.pdf -- Also there's Schuam's Abstract Algebra exercise 337, a diagram is shown on arxana.net. - - Redone using IATC: http://metameso.org/ar/schuams-example.pdf -- An example from the Ganesalingam and Gowers paper: http://metameso.org/ar/robotone-example.pdf -- Galois theory. Emil Artin's book. -- Euclid algorithm -- possibly an example from metamath or coq or whatever to show that we can model that stuff as discourse... -- ... - -*** Other Background - -**** Conference and workshop "due dilligence" - -Previous publications in the ICFP series touch on themes below (i.e., - this a selection of papers from the proceedings over the last 10 - years that mention "mathematics" or something similar; hopefully at - least a few of them are relevant). We plan to submit to the - associated [[http://www.schemeworkshop.org/][Scheme and Functional Programming]] workshop, though perhaps - we might also want to have a look at the [[http://functional-art.org/2017/cfp.html][Functional Art, Music, - Modelling and Design]] workshop in case it proves to be a better fit. - -- /A functional programmer's guide to homotopy type theory/, -- /Program synthesis: opportunities for the next decade/, -- /Lem: reusable engineering of real-world semantics/ [Lem stands for "[[http://www.cl.cam.ac.uk/~pes20/lem/][Lightweight executable mathematics]]"], -- /Functional programming with structured graphs/, -- /Painless programming combining reduction and search: design principles for embedding decision procedures in high-level languages/, -- /Using camlp4 for presenting dynamic mathematics on the web: DynaMoW, an OCaml language extension for the run-time generation of mathematical contents and their presentation on the web/, -- /TeachScheme!: a checkpoint/, -- /A functional I/O system or, fun for freshman kids/ and -- /Commutative monads, diagrams and knots/ - -**** Items we've worked on before - -- iatc :: The primary features of the adaptation of IAT to IATC - are: - -1. to introduce more explicit relationships between pieces of -mathematical content; and -2. to advance a range of ``predicates'' that relate mathematical -propositions. - -These predicates describe inferential structure, judgements of -validity or usefulness, and reasoning tactics. - -- scholia :: The kind of collaborative online - knowledge-building that happens on sites like - Wikipedia \cite{BENKLER2002} (or in other digital - libraries broadly construed \cite{KROWNE2003}). - One familiar class of examples is provided by - revision control systems, e.g., Git, which builds - a network history of changes to documents in a - directed acyclic graph (DAG). - -- arxana :: We have been specifically interested in - applications within mathematics, which has given - Arxana a unique flavour. In particular, we propose - to use a scholium model to model *the logic of - proofs*, not just to represent mathematical texts - for reading. Over the years as we've worked on - Arxana, we have generated all sorts of files. Many - of these are stored in - =/home/shared-folder/public_html/=, and accessible - on the web at [[http://metameso.org/ar/]]. Earlier - prototypes are available via http://arxana.net. - -***** Research notes - -- "nemas" document: http://metameso.org/ar/metamathematics.pdf -- Joe's thesis Appendix B: http://metameso.org/ar/thesis-appendix-b.pdf -- Metamathematics preprint: http://metameso.org/ar/mathematical-semantics.pdf -- hcode function spec: http://metameso.org/ar/hcode.pdf -- Lakatos game spec: http://metameso.org/ar/lakatos-flowchart.pdf - -***** Wiki - -- [[http://metameso.org/cgi-bin/current.pl?action=browse;oldid=Arxana;id=A_scholium-based_document_model][A scholium-based document model]] is the wiki homepage for Arxana and has several relevant discussions. -- [[http://metameso.org/cgi-bin/current.pl/Two_years_later][Two years later]] collects many of the mathematical themes in - overview -- [[http://metameso.org/cgi-bin/current.pl/samples_of_formal_mathematical_writing][samples of formal mathematical writing]] is a small collection - of statements in mathematical language (leaving out ``the part - in between dollar signs'' - -***** Website - -- [[http://arxana.net][arxana.net]] has links to source code from various prototypes - - It also has a small gallery of hand-drawn diagrams - -**** And other things elsewhere: - -- /Conceptual dependency theory/ has been used for such purposes, e.g., -in the context of machine understanding of stories \cite{LEON2011}. -- Zooming in and out to find the right level of detail is related to Faré's categorical theory of levels of a computer system [add citation]. -- /Handbook of artificial intelligence/ by Barr and Feigenbaum describes *SAM* and *PAM*, the /Script Applier Mechanism/ and /Plan Applier Mechanism/ by Schank and Abelson et al., on page 306 et seq. \cite{barr1981handbook} -- Winograd, Natural language and stories --- MIT AI Technical Report 235, February 1971 with the title "Procedures as a Representation for Data in a Computer Program for Understanding Natural Language" ([[http://hci.stanford.edu/winograd/shrdlu/AITR-235.pdf][link]]) - - Could we reuse this stuff for the "narrative" - - How do we know how to read the text and transform it into IATC? - - An example is Gowers's "total stuckness": how do we turn that into code? -- Conceptual dependency representation - from "Deep Blue" website ([[https://deepblue.lib.umich.edu/bitstream/handle/2027.42/30278/0000679.pdf?sequence=1][link]]). - - Schank's MARGIE system is described on page 300 /et seq./ of \cite{barr1981handbook}. -- AI textbook in wiki. - - -** Proofs and prologues - -In several of our examples, such as the Gowers problem and Artin's -book, on sees a structure in which a proof or calculation is preceded -by a kind of prologue in which the author describes motivation and -strategy. Unlike the proof proper which consists of mathematical -statements and deductions, the language here is more general and -the reasoning may be abductive, inductive, or heuristic. - -We would like to have a representation for these sorts of prologues -and indicate how they connect with the proof which they introduce -as well as the rest of the text. A challenge here is that the -texts in question can allude and refer to all sorts of referents -including themselves at a high level of abstraction and require -a certain amount of common sense to be understood. - -Since the issues that arise here are similar to those which arise in -undestanding and representing natural language stories, we should be -able to adapt techniques such as conceptual dependency relation or -goals and themes to a mathematical context. - - -*** TODO We should figure out what the inferences between basic assertions (I/R, I/E, etc.) would look like here. (e.g. fig. 7 of Lytinen) - -- links between basic CD's - - e.g., joe communicated the IP address to ray, by talking, so that ray could get on the server - -** Goals - -In order to figure out why and how actions take place, CD theory -introduces procedural knowledge in the form of scripts, MOPs, TOPs, -goals, and plans. For our purposes, goals and plans will be used. - -This relates to the =meta= items of the spec. Maybe we should -expand the =goal= item using some sort of ontology like that of -Schank and Ableson. - -** Parsing - -We saw earlier how Lytinen schematizes the sentence "John gave Mary a -book" as an s-expression. - - -*** TODO Initial non-working pseudocode prototype - -Here it makes sense to at least try and transcribe the main examples -in some "pseudocode", inspired by IATC and hcode, which we can then -refine into a working version later. - -Several examples have been transcribed: - -- http://metameso.org/ar/mpm3.html -- http://metameso.org/ar/gowers2012.html -- http://metameso.org/ar/robotone-opensets.html - -However, as per the headline, these don't currently *do* anything. - -*** TODO Basic working version - -*** TODO Can the system come up with answers to new, basic, questions? - -- Inspired by Nuamah et al's geography examples -- Simple comparisons, like, how does this concept relate to that concept? We informally talk about this as ``analogy'' between concepts. But... - -*** TODO Foldable views (like in Org mode) so that people can browse proofs - -- This may come after the May submission -- Folding and unfolding the definitions of terms in something like an APM context is relevant example. Just `unpacking' terms. - - -#+C Local Variables: -#+C org-tree-slide-skip-outline-level: 3 -#+C mode-line-format: nil -#+C eval: (org-display-inline-images t t) -#+C eval: (setq org-tree-slide-header nil) -#+C End: - - -* Frame with references :noexport: - :PROPERTIES: - :BEAMER_OPT: fragile,allowframebreaks,label= - :END: - \printbibliography +#+TITLE: Extending the LISP model: from cons cells to triples, from trees to hypergraphs +#+AUTHOR: Joseph Corneli and Raymond Puzio +#+EMAIL: contact@planetmath.org +#+DATE: \Lightning[fill=yellow] Saturday, 3 September, 2017 (Scheme@ICFP) \Lightning[fill=yellow] +#+DESCRIPTION: Organizer for presentation on arxana and math text analysis at Oxford. +#+KEYWORDS: arxana, hypertext, inference anchoring +#+LANGUAGE: en +#+OPTIONS: H:1 num:t toc:nil \n:nil @:t ::t |:t ^:nil -:t f:t *:t <:t +#+OPTIONS: TeX:t LaTeX:nil skip:nil d:nil todo:t pri:nil tags:not-in-toc +#+INFOJS_OPT: view:nil toc:nil ltoc:t mouse:underline buttons:0 path:http://orgmode.org/org-info.js +#+EXPORT_SELECT_TAGS: export +#+EXPORT_EXCLUDE_TAGS: noexport +#+LATEX_CLASS: beamer +#+LATEX_CLASS_OPTIONS: [presentation,serif] +#+LATEX_HEADER: \usefonttheme{professionalfonts} +#+LATEX_HEADER: \usepackage{fontspec} +#+LATEX_HEADER: \setromanfont{TeX Gyre Pagella} +#+LATEX_HEADER: \usepackage{mathtools} +#+LATEX_HEADER: \usepackage{unicode-math} +#+LATEX_HEADER: \setmathfont{TeX Gyre Pagella Math} +#+LATEX_HEADER: \setmonofont[Color=blue]{TeX Gyre Adventor} +#+LATEX_HEADER: \usepackage{listings} +#+LATEX_HEADER: \usepackage{parskip} +#+LATEX_HEADER: \newfontfamily{\lstsansserif}[Scale=.85]{TeX Gyre Adventor} +#+LATEX_HEADER: \lstset{basicstyle=\lstsansserif\color{blue},keywordstyle=\lstsansserif\color{blue}} +#+LATEX_HEADER: \definecolor{fondpaille}{cmyk}{0,0,0.1,0} +#+LATEX_HEADER: \definecolor{sepia}{rgb}{0.44, 0.26, 0.08} +#+LATEX_HEADER: \setbeamercolor{background canvas}{bg=fondpaille} +#+LATEX_HEADER: \setbeamercolor{normal text}{fg=sepia} +#+LATEX_HEADER: \beamertemplatenavigationsymbolsempty +#+LATEX_HEADER: \usepackage{lightning} +#+LINK_UP: +#+LINK_HOME: +#+HTML_HEAD: +#+STARTUP: showeverything + +# #+LATEX_HEADER: \setbeamertemplate{footline}[frame number] + +# * Title slide + +# Extending the LISP model: +# from cons cells to triples, from trees to hypergraphs + +# Joseph Corneli and Raymond Puzio + +# Saturday, 3 September, 2017 +# Scheme@ICFP +# Oxford, UK +* From cons cells to triples + +*Arxana* uses a higher-dimensional variant of LISP, based on nested +semantic networks instead of cons cells. + +In contradistinction to LISP where the fundamental building block is a +cell =(a . b)=, Arxana's fundamental building block is a triple, =(a c +b)=. + +Instead of =car= and =cdr=, we have =src=, =txt=, and =snk=. + + +* From trees to hypergraphs + +Furthermore, in the language of the Semantic Web, every triple is +`reified'. Links and their contents can be augmented with offset +annotations, and can contain further structure. + +Pierre De Lacaze shared an illustrative example of the kind of +structure we're talking about, and why it is different from more +elementary relations in a semantic network. + +/Mom resents the fact that John disapproves of Jane and Jim's +marriage./ + +#+ATTR_ORG: :width 400 +#+ATTR_LATEX: :width 5cm + [[file:./mom.png]] + +* Illustration of basic mechanics + +Both nodes and links are represented as triples, and we collectively +call them /nemas/. A repository of nemas is a /plexus/. + +1. Each plexus has a distinguished element =0=, called *ground*. + +2. If we set =α:=(A D E)=, then =(α C B)= is an assertion about the + link =(A D E)=. + +3. On the other hand, =((get-src α) D E)= is about the =src= of =α=, + which happens to be =a= at present (but nemas are mutable). + +4. If we wanted to make an assertion about =A= "itself", we can + represent it as a nema with the special form =β:=(0 A + 0)=. Subsequently, =(β C B)= is an assertion about =A=. + +[That said, =β= would still be mutable, is this a problem? -JC] +* What do we get? + +These facilities allow us to build, reason about, query, and program +with hypergraphs rather than trees or networks. + +Unlike mainstream graph databases, code can be stored on links. This +representation strategy is useful for building runnable conceptual +models of complex and recursive structures. + +In particular, we have found this representation strategy well-suited +to modelling the non-deductive style of informal mathematical +dialogues and expositions. + +* Related work + +Historical programming languages which support a similar annotative +style include Kurzweil el al's /Flare/ and Nelson's /ZigZag/. + +A more recent endeavour is /AtomSpace/, which is a more mature piece +of software than Arxana, e.g., it comes with both Scheme and Python +bindings and a range of optimisations. + +The simplicity of Arxana makes it useful for prototyping! + +Also, it is implemented in Emacs Lisp, which we have found useful (in +particular, stable) for our sub-part-time workflow. + +* History of development and outreach + +- Free Culture and the Digital Library Symposium, Atlanta, 2005 +- Experiments with a Common Lisp + cl-elephant backend, 2007 +- Rewrite with a Common Lisp + SQL backend, 2010 +- Rewrite with the storage model described here, 2013 +- Presentations at LispNYC, 2005 and 2013 +- Presentation at the Emacs Conference, London, 2013 +- Presentation at the monthly Emacs Meetup, NYC, 2017 +- Next Saturday: presentation at FARM 2017 + +* Other relevant work: Conceptual Dependence theory + +E.g., "John gave Mary a book." +#+ATTR_ORG: :width 800 +#+ATTR_LATEX: :width 11cm + [[file:./john-example.png]] +#+BEGIN_SRC lisp +(atrans actor (person name john) + object (phys-obj type book) + recip (person name mary)) +#+END_SRC + +\bigskip + +*NB. We view Conceptual Dependence /primitives/ (like =atrans=) as +analogous to /performatives/ (like =Assert=, =Challenge=, etc.).* + + +* Illustration: Application to mathematics +``I have seen this problem, that if $G$ is a finite group and $H$ is a +proper subgroup of $G$ with finite index then $ G \neq +\bigcup\limits_{g \in G} ghg^{-1}$. Does this remain true for the infinite case also.'' +#+ATTR_ORG: :width 800 +#+ATTR_LATEX: :width 10cm + [[file:./example-graph.png]] +*``There's something I don't understand here: do you perhaps mean +$gHg^{-1}$ instead of $ghg^{-1}$?''* + +* Input: +For example, the first assertion above: +#+BEGIN_SRC lisp +(read-tree '(Assert + (implies + (conjunction (finite-group G) + (subgroup H G) + (has_property (index H G) + finite)) + (not (equal G + (union (times g + H + (inverse g)) + :over g :in G)))))) +#+END_SRC + +* Output: + +Nodes are given unique (=gensym='ed) labels; nested code is not. + +#+BEGIN_SRC lisp +(Assert123 + (implies124 + (conjunction125 (finite-group G) + (subgroup H G) + (has_property126 (index H G) + finite)) + (not127 (equal G (union (times g + H + (inverse g)) + :over g :in G))))) +#+END_SRC + +The labels can then be referred to in subsequent input, so that +additional structure can be added relative to existing structure. + +* The end + +That's all for now! + +- =joseph.corneli@ed.ac.uk= +- =rspuzio@planetmath.info= + +PS. Maybe see you on our "mob" branch. + +- http://repo.or.cz/w/arxana.git + +* outtakes :noexport: + +** Other things we've worked on + +- "nemas" document: http://metameso.org/ar/metamathematics.pdf +- Joe's thesis Appendix B: http://metameso.org/ar/thesis-appendix-b.pdf +- Metamathematics preprint: http://metameso.org/ar/mathematical-semantics.pdf +- hcode function spec: http://metameso.org/ar/hcode.pdf +- Lakatos game spec: http://metameso.org/ar/lakatos-flowchart.pdf +- [[http://metameso.org/cgi-bin/current.pl?action=browse;oldid=Arxana;id=A_scholium-based_document_model][A scholium-based document model]] is the wiki homepage for Arxana and has several relevant discussions. +- [[http://metameso.org/cgi-bin/current.pl/Two_years_later][Two years later]] collects many of the mathematical themes in + overview +- [[http://metameso.org/cgi-bin/current.pl/samples_of_formal_mathematical_writing][samples of formal mathematical writing]] is a small collection + of statements in mathematical language (leaving out ``the part + in between dollar signs'') + +#+C Local Variables: +#+C org-tree-slide-skip-outline-level: 3 +#+C mode-line-format: nil +#+C org-latex-compiler: "xelatex" +#+C eval: (org-display-inline-images t t) +#+C eval: (setq org-tree-slide-header nil) +#+C End: + +* Frame with references :noexport: + :PROPERTIES: + :BEAMER_OPT: fragile,allowframebreaks,label= + :END: + \printbibliography -- 2.11.4.GIT